Application Of TribologyTribology of gears In this topic, we shall discuss about the geometry
of gears and then consider three different aspects of gearing namely elastohydrodynamic
lubrication, tooth contact phenomena, and wear of gears. Gears are machine elements, which are
required to transmit power between shafts rotating at different rotational speeds. By adding teeth
of the proper shape on disk, power can be transmitted without slip at uniform rate. These types of
geometrics are known as external gears. Internal gears are generally more efficient since the
sliding velocity along the profile is lower than equivalent external gears. It operates at closer
center distance with its mating pinion than external gears of the same size, which often permits a
more compact design. The internal gears eliminates the use of an idler gear, where it is necessary
to have two parallel shafts rotate in the same direction. In manufacturing point of view also,
external gears are simpler than internal gears.
Fig. 6.52: Internal & External gearing Often gears are treated as pitch cylinder, which roll
together without slip as shown in Fig. 6.53. Smaller gear is known as pinion and larger mating
gear is called gear. Generally, a gear pair acts as a speed reducer aiming torque amplification at
output shaft.
Fig. 6.53: Gear action . Tooth Profile : Generally tooth profile is designed so that velocity ratio
does not change due to inaccuracies in center distance. Involute profile may be visualized as the
locus of points generated by the end of a string, which is held in tension as it is wounded from a
drum/cylinder, as shown in Fig.
6.54.
Fig. 6.54: Concept of involute profile. Tooth curves of the mating teeth need to be tangent to
each other as shown in Fig. 6.55. Line of action is tangent to both pinion & gear base circles. On
changing center distance, line of action still remains tangent to both base circles but slope
changes.
Fig. 6.55: Gears in action [1]. Pressure angle: Pressure angle is expressed as Φ1 = cos1(R
b/RI).
Nominal pressure angle is given by :
The lower pressure angle has the advantage of smoother and quieter tooth action because of
larger profile contact ratio. In addition, lower loads are imposed on the support bearings because
of a decrease radial load component. Pressure angle at base circle is
zero.
Backlash : Difference between tooth space and tooth
thickness is known as backlash. It prevents jamming of teeth and compensates for thermal
expansion of teeth.
Fig. 6.56: Backlash [1]. Velocity ratio : The amount of speed reduction is simply the ratio of
pitch diameters of the larger gear to the smaller gear. There is no limit to the speed reduction
ratio that can be achieved using gearing; but larger ratio must be obtained using multi-stage
reduction. In simple gear mesh, a maximum ratio in order of 7:1 to 10:1 should not be exceeded.
The limit on velocity ratio depends on gear pair for example :
Speed
•
reduction
Spur
for
a
single
gear
<
pair
of
7:1
•
Helical
•
Internal
4-8
•
Bevel
1-8
•
Cylindrical
10:1
worm
3-80.
For high speed reduction, two stage or three stage
construction are preferred, otherwise gear wheel size Fig. 6.57: Compound gear
increases, which increases the gearbox size. For high
train.
speed reduction, compound gear trains are required.
In such trains(as shown in Fig. 6.57), at least one
shaft carries two gears.
Fig. 6.58(a): Spur gear & Fig. 6.58(b): Helical gear. An efficient method of achieving high
reduction ratios in minimum space is the use of planetary gearing. Helical gearing, in which the
teeth are cut at an angle with the axis of rotation, was developed after spur gearing and has the
advantages that has the smoother action and tends to be quieter(Fig. 6.58b). In helical angle
greater than 15 degrees, the tooth bending capacity generally begins to drop off due to the fact
that the tooth thickness decreases rapidly. In addition, helical gears causes axial thrust force and
impose load on bearings. Rolling and sliding in gears :
Ideally, rolling gears are required. In practice, sliding comes along rolling action. and therefore,
lubrication of gears is required. Typical gear pair having high sliding is shown in Fig. 6.59 to
Fig. 6.61.
Fig. 6.59: Bevel gears
Fig. 6.60: Helical crossed axis gears.
Fig. 6.61: Worm gears [1].
Worm gears have crossed axes(Fig. 6.61), line contact and a very large sliding component.
Helical crossed axes(Fig. 6.60) have point contact and large sliding component. Hypoid
gears(Fig. 6.59) are offset bevel gears. Hypoid gears are widely used in many power trains to
transfer power between two non-intersecting crossed axes. Their most common and highestvolume applications can be found in front and rear axles of rear-wheel-drive or all-wheel-drive
vehicles. It is interseting to note that even spur gears experience sliding ,as depicted in Fig.
6.62.
Fig. 6.62: Sliding action in spur gears. For rolling action, tangential velocities at point of
contact must be equal to make sliding zero. This happens at pitch point. At all other contact
points(as shown in Fig. 6.63 at point I), radius of gear and pinion will change (rotational speed
remain constant) and that will introduce sliding.
Difference between
VPI and VGI will provide positive/negative sliding
speed.
Fig. 6.63: Sliding at point I Friction & Lubrication Of Gears Friction between gear pair occurs
due to sliding between meshed teeth and churning of lubricant. In absence of lubricant additives
and antifriction coating, gears will be subjected to direct friction. This is hypothetical situation
which occurs rarely in extreme conditions. A worst situation from tribology point of view i.e.
high temperature, very high load, etc. In normal atmospheric conditions, all engineering metallic
surfaces are primarily coated with some adsorbed gas(Fig. 6.64) and/or fluid films(Fig. 6.65).
Shear strength of coating(τi) is lower than that of the base material(τy) as shown in following
equation, they are continually rubbed off and reformed. They thus protect the surface of the base
material from excessive wear and subsequent destruction. This favorable behavior is utilized by
intentionally creating protective surface coating. Following two coatings(as solid lubricants) are
used to reduce
friction.
Fig. 6.64: Gear surface with adsorbed gases. • Phosphate layers, few microns.
• Graphite or molybdenum disulphide, 1-2 μ thin
coating.
Fig. 6.65: Gear surfaces coated with boundary additive or antifriction coating. Mixed
Friction : This friction process is aided by the presence of small quantities of lubricant, at the
point of friction. This friction process is characterized by solid-body friction as well as by fluid
friction(Fig. 6.66), therefore, it is called mixed friction. In this regime, friction and wear are
influenced by the ability of lubricant to create protective boundary films on gear tooth with
chemical and physical reactions, and by its viscous characteristics. Interface shear strength in
mixed lubrication can be given
by;
Fig. 6.66: Mixed lubrication.
Fig. 6.67: Oils samples. It is interesting to note that oil sample collected after 3 hours of
operating gear at 500 rpm (no load conition) show wear(Fig. 6.67). This means mild wear is
bound to occur in gear operations. Power Loss : With involute profile of gears, only one contact
position experiences pure rolling. As contact moves towards or away from pitch point, sliding
occurs. Due to sliding, power loss occurs and transmission efficiency decreases. Typical values
of gear efficiencies are listed in Table 6.16. Table 6.16: Gear efficiency.
Toothing load losses certainly account for most
of the losses. Toothing Power Loss : In a geared system, the total power loss is comprised of two
groups of losses: (i) load-dependent (friction induced) mechanical power losses and (ii) loadindependent (viscous) spin losses. Sliding and rolling friction losses at the loaded gear meshes
and at the bearings largely define the load-dependent mechanical power losses. The total
mechanical loss is then given as the sum of losses from all gear meshes and bearings. The sliding
friction losses are related to the coefficient of friction, normal load and sliding velocity on the
contact surfaces, while the rolling friction losses occur due to the formation of an
elastohydrodynamic (EHL) film. It is interesting to note that coefficient of friction is variable
and it depends on operating conditions. Friction losses can be divided into two major categories:
Loss due to load and loss due to speed. In high speed units, the churning losses may exceed the
friction losses; therefore, the type and amount of lubricant are critical. • Load power loss.
Pload = Fr vg
Pload = μ Wn vg Normal gear load (Wn) for a given application depends on pitch diameter and
face width. These dimensions are determined on the basis of tooth stresses, which are imposed
by the transmitted tooth load. The tooth load is simply the torque on a given gear divided by the
gear pitch radius. Torque is calculated from the horsepower transmitted and the speed of rotating
component in question.
Fig. 6.68: Normal load on tooth.
Coefficient of Friction : μm = f(Wn, b, Ve, η, Rcomposite)
"Entraining velocity" - summation of rolling
velocities.
Fig. 6.69: Higher entraining velocity increase "spin losses". Composite roughness depends on
gear manufaturing process as given in Table 6.17. The dynamic viscosity η(νP) depends on
operating temperature. The Walther`s relation relating kinematic viscosity(ν) and absolute
temperature is given by; Table 6.17: Rcomposite.
log10 log10 = (ν + 0.8) =
Alog10T + B.
Gear life depends on effective lubrication, which can be quatified by minimum film thickness to
Rcomposite, as shown in Fig.
6.70.
Fig. 6.70: Dependence of gear life on film thickness/surface roughness ratio. Lubricant Film
Thickness : Curve fit EHL equation for minimum film thickness is given
as;
Estimating effective temperature
: Following empirical formulae are used to estimate effective
temperature(TF).
Fig. 6.70: Contact stress. where, σ is contact stress, V1 is tangential velocity of pinion, V2 is
tangential velocity of gear and w is width of contact patch as shown in Fig. 6.70.
Coefficient of thermal contact, β = thermal conduction * specific heat * density.
β = λCρ; On substituting value of contact patch, 2b and pmax from
equation(12).
where
...........
Assuming same materials for gear &
pinion.
where parameter ƒZ is decided based on number
of teeth on pinion as given in Table 6.18 : Table 6.18: Parameter ƒZ.
Higher value of Tf cause scuffing failure of gears. To reduce the
value of Tf, effective lubrication that maintains low friction coefficient is desirable. But
sometime failure of lubrication (pump failure, filter chocking, excessive leakage) occurs and gear
materials must be able to handle such extreme situations. To understand this, let us consider two
unlubricated gear pairs, one of Nylon/Nylon and other Nylon steel pair. Nylon/Nylon gear pair
friction is slightly more than Nylon/Steel gear pair.
Given data: Z = 17, m = 3 mm, b = 30 mm, ω = 150 rad/s, power = 850 W. Table 6.19: Material
parameters.
By using
parameters shown in Table 6.19, calculate the value of temperature. For the N-N temperature is
1610C but for N-S temperature is 19.50C ,which is very less value compared to the N-N gear pair
temperature. So N-S is prefferable.But for home applications, we use N-N gear pair, where rate
of heat generation is relatively low. Surface fatigue of spur gears To estimate the working life of
gears, it is essential to analyze the destructive forces at work, and knowledge of the ability of
chosen gear materials to withstand those forces. Fig. 6.71 shows surface pitting and bending
failure of gear tooth. To design gear, we need to estimate Wn (Fig. 6.71) and corresponding
contact stresses. Bending & contact stresses must be within modified Goodman line for material.
Properly designed gear-sets should never fail but must be expected to eventually fail by wear of
one of surface. Insufficient backlash is sometimes the cause of excessive heat and wear. If
sufficient backlash has not been provided to take care of the differential thermal expansion, the
teeth will bind, with disastrous results. Inadequate lubrication may also be a source of excessive
heat and wear.
Fig. 6.71: Failures of gear. Wear can be approximate using Archard`s equation.
Wear volume = [(K1 * load * sliding velocity)/(3 * hardness)]
Lesser value of K1 (i.e, better lubrication), lower load and high hardness reduce wear and
enhance gear life. Surface/Contact stresses in spur gears : Surface failure of gear tooth occurs
due to very high local contact stresses. Maximum contact pressure at the contact point between
two cylinders is given by :
12 where
pmax is max contact stress and b is half of contact patch.
d1 and d2 are curvatures of the profile at the point of contact.
On substituting, W = Wt/cos Φ, d1 = dp * sin Φ, L =
F.
On substituting expression of b
in pmax.
Maximum contact stress
is equal to pmax. Therefore, contact stress
is;
rearranging;
On
Table 6.20:
Correction factors.
As
per AGMA, we must include application factor, Ca (Table 6.20), load distribution factor,
Cm (table 6.20) and velocity factor, Cv to estimate contact stress. Calculation of Factor Cν
: Factor Cy depends on velocity and gear quality.
Table
6.21: Qv vs geometric tolerance.
Example : A gear pair (ZP=23, θ = 200, Zg = 24, m =
1.75, F = 10.0 mm) transmits 8 N.m torque from crankshaft (rotational speed 8000 rpm) of single
cylinder IC engine to wheels. Bore diameter of pinion is 17 mm, and bore dia of gear is 20 mm.
Using AGMA pitting equation formula, determine the maximum contact stress. Assume gears’
quality = 9, E = 2.e5 MPa, μ =
0.3
To avoid gear
failure, σC must be lesser than material strength. Contact Stress vs. Brinell Hardness
:
Fig. 6.72: Effect of Brinell hardness on allowable contact stress for through-hardness steel
[2]. The Fig. 6.72 shows that strength of a gear tooth is proportional to the hardness of the steel.
Most gears are in the hardness range of approximately Rc 30 to Rc 38 or Rc 55 to Rc 64. The
region from Rc 30 to 38 is usually termed as “through hardened”, while the range Rc 55 to 64 is
almost always “surface hardened, where the tooth has a hard surface case and a softer inner
core. Lubrication : Too much or too less lubricant is harmful for gear operation. Usually,
following two types of lubrication mechanisms are commonly used for gear lubrication.
• Splash lubrication, when power transmitted < 100 kW and Pitch_vel < 10 m/s.
• Pressurized lubrication (by oil jets) for large gear train transmitting power greater than 100
kW.
Following empirical formulae are availiable for splash lubrication system to minimize churning
losses.
Recommended oil viscosity [2].
Table 6.22:
With increase in pitch line
velocity, lubricant used should be less viscous in order to min. power losses. In case of heavily
loaded gears, however, more viscous lubricant will be recommended. Table 6.22 provides an
initial guidance for lubricant selection. Long service life, free from wear problems depends on
lubrication system; its ability to keep gear cool, and to deliver lubricant free from hard particles
(filter with 5 micron rating). Lack of lubricant, may initiate scuffing failure. References : 1.
Hamrock B J, Jacobson B O & Schmid S R, Fundamentals of Machine Elements, McGraw-Hill
Inc., 1998.
2. Shigley J E, Mischke C R, Mechanical Engineering Design, Tata McGraw-Hill Publishing
Company Limited, 2003.